Minuscule Coxeter Dressians

In this extended abstract, we study special tropical prevarieties which we call Coxeter Dressians. They arise from equations capturing a generalization of valuated symmetric basis exchange for Coxeter matroids. In particular, we study subdivisions of the associated Coxeter matroid polytopes. We show that the subdivisions induced by points of the Coxeter Dressian consist of cells which are strong Coxeter matroidal. This generalizes well-known results in type $A$ to other Lie types. Finally, we implement explicit computations of Coxeter Dressians in OSCAR.

💡 Research Summary

This extended abstract presents a systematic study at the intersection of tropical geometry and Coxeter matroid theory, introducing and analyzing a new class of objects called Coxeter Dressians.

The research generalizes the well-established framework for type A_n matroids (ordinary matroids) to other Lie types (B, C, D, E6, E7). The starting point is the strong exchange property for Coxeter matroids, a set of subsets of the coset space W/P (for a Weyl group W and a parabolic subgroup P) that generalizes the basis exchange axiom of ordinary matroids. When P is minuscule, these sets correspond to points on a generalized flag variety, defined by a system of quadratic strong exchange equations.

The authors define the Coxeter Dressian Dr(W, P) as the tropical prevariety obtained by tropicalizing these strong exchange equations. A point in the Dressian is called a valuated Coxeter matroid.

The paper’s first major result (Theorem A) states that for any minuscule type, a point μ in the Coxeter Dressian induces a regular polyhedral subdivision of the associated Coxeter matroid polytope (the convex hull of its support), and furthermore, every cell in this subdivision is itself the polytope of a strong Coxeter matroid. This generalizes the fundamental result that points in the classical Dressian (type A) induce matroidal subdivisions of the hypersimplex. The proof hinges on showing the strong exchange equations are affinely invariant.

The second key result (Theorem B) provides a simplification for type B_n. It shows that for a function μ whose support is a strong Coxeter matroid, membership in the Coxeter Dressian is equivalent to satisfying only a specific subset of 4-term strong exchange equations, analogous to how 3-term Plücker relations suffice in type A.

Beyond theory, the paper emphasizes computational exploration. Using the computer algebra system OSCAR, the authors implement algorithms to compute Coxeter Dressians explicitly for small cases in types B_n (n=3,4) and D_n (n=5,6). They analyze their structure, compare them to secondary fans, and compute f-vectors. Notably, they provide a counterexample (Example 4.2) showing that the converse of Theorem A is false in type B_3: not every height function inducing a strong matroidal subdivision lies in the Dressian. They also perform exhaustive computational verification that the strong exchange property holds for all minuscule Coxeter matroids of type E_7.

In summary, this work successfully extends the tropical geometry of matroids to the broader context of Coxeter groups, establishing foundational theorems about the associated Dressians and supporting them with both theoretical proofs and concrete computational evidence.